Mikhail Shaposhnikov, Christof Wetterich
(Submitted on 1 Dec 2009 (v1), last revised 12 Jan 2010 (this version, v2))
There are indications that gravity is asymptotically safe. The Standard Model (SM) plus gravity could be valid up to arbitrarily high energies. Supposing that this is indeed the case and assuming that there are no intermediate energy scales between the Fermi and Planck scales we address the question of whether the mass of the Higgs boson $m_H$ can be predicted. For a positive gravity induced anomalous dimension $A_\lambda>0$ the running of the quartic scalar self interaction $\lambda$ at scales beyond the Planck mass is determined by a fixed point at zero. This results in $m_H=m_{\rm min}=126$ GeV, with only a few GeV uncertainty. This prediction is independent of the details of the short distance running and holds for a wide class of extensions of the SM as well. For $A_\lambda <0$ one finds $m_H$ in the interval $m_{\rm min}< m_H < m_{\rm max}\simeq 174$ GeV, now sensitive to $A_\lambda$ and other properties of the short distance running. The case $A_\lambda>0$ is favored by explicit computations existing in the literature.